The universality class of diffusion-limited aggregation and viscous fingering

– We investigate whether fractal viscous fingering and diffusion-limited aggregates are in the same scaling universality class. We bring together the largest available viscous fingering patterns and a novel technique for obtaining the conformal map from the unit circle to an arbitrary singly connected domain in two dimensions. These two Laplacian fractals appear different to the eye; in addition, viscous fingering is grown in parallel and the aggregates by a serial algorithm. Nevertheless, the data strongly indicate that these two fractal growth patterns are in the same universality class. Laplacian fractals are paradigmatic examples for the spontaneous growth of fractal patterns in natural systems. In particular, two such examples have attracted an enormous amount of interest: viscous fingering and diffusion-limited aggregation (DLA). Viscous fingering is realized [1] when a less viscous fluid displaces a more viscous fluid contained in the narrow gap between two glass plates (a Hele-Shaw cell). When the less viscous fluid is inserted through an aperture in one of the glass plates, a pattern is formed, as shown in fig. 1 [2]. The fluid velocity v of the displaced fluid in a Hele-Shaw cell satisfies Darcy’s law, v ∝ ∇p, where p is the pressure. To a very good approximation, the viscous fluid is incompressible (i.e., ∇ ·v = 0), so the pressure in the viscous fluid satisfies the Laplace equation ∇2p = 0. DLA is realized [3, 4] as a computer experiment in which a fractal cluster is grown by realising a fixed-size random walker from infinity, allowing it to walk until it hits any particle already belonging to the cluster. Since the particles are released one by one and may take a long time to hit the cluster, the probability field is stationary and, in the complement of the cluster, we have again ∇2p = 0. A typical DLA cluster is shown in the right panel of fig. 1. Mathematically, these problems are similar but not identical. In both cases, one solves the Laplace equation with the same boundary condition at infinity, i.e. ∇p = const × r̂/r as r → ∞. However, in viscous fingering, each point on the fractal’s boundary advances at c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2006-10246-x 258 EUROPHYSICS LETTERS Fig. 1 – Left panel: digitized image of an experimental viscous fingering pattern. Air (black) is injected into the oil-filled gap (white). The pattern is approximately 22 cm in diameter, cf. [2]. Right panel, image courtesy of Ellak Somfai: a diffusion-limited aggregate with 50000000 particles. a rate proportional to ∇p, whereas the DLA accretes one particle at a time, changing the Laplacian field after each such growth step. Thus, we refer to viscous fingering and DLA as parallel and serial processes, respectively. In addition, the ultraviolet regularization differs; in DLA, p = 0 on the cluster and the regularization is provided by the particle size. In viscous fingering, one solves the problem with the boundary condition p = σκ, where σ is the surface tension and κ is the local curvature. Finally, viscous fingers are grown in a finite gap and are not truly 2-dimensional. Accordingly, one can ask whether these two fractal growth problems are in the same scaling universality class. But to answer this question, one must first define what one means by a “scaling universality class.” Definition of the scaling universality class. having two fractal growth patterns, deciding whether they belong to the same universality class involves comparing both their geometry and their growth dynamics. The geometric correspondence between the fractal patterns may be answered by measuring their fractal dimension D0. Denote by Rn the radius of the minimal circle that contains a fractal pattern. The fractal dimension is defined by how the mass Mn contained within this circle (number of particles for DLA, area for viscous fingers) scales with Rn, Mn ∼ R0 n . Measurements of this type indicated a value D0 ≈ 1.71 for both problems, motivating many authors to express the opinion that these two problems are in the same universality class [2,5]. Obviously, the fractal dimension by itself is not sufficient to determine the growth dynamics, and a more stringent definition of a universality class is necessary. We propose here that the identity of the scaling properties of the harmonic measure is a sufficient test for two Laplacian growth problems to be in the same universality class. The harmonic measure is the probability for a random walker to hit the boundary of the fractal pattern. It determines the growth in both problems, being proportional to ∇p at the boundary. Suppose that we know the probability measure μ(s)ds for a random walker to hit an infinitesimal arclength on the fractal boundary. We compute the probability Pi( ) ≡ ∫ i-th box μ(s)ds, and then define the generalized dimensions [6] via Dq ≡ lim →0 log ∑N( ) i=1 P q i ( ) (q − 1) log , (1) J. Mathiesen et al.: The universality class etc. 259 where N( ) is the number of boxes of size that cover the fractal boundary. The q > 0 branch of this function probes the high-probability region of the measure, whereas the q < 0 branch stresses the low probabilities. For q → 0 we find the fractal dimension D0. An equivalently useful description [7] is provided by the scaling indices α that determine how the measure becomes singular, Pi( ) ∼ , as → 0. This set of indices is accompanied by f(α) which is the fractal dimension of the subset of singularities of scaling exponent α. The relation of these objects to Dq is in the form of the Legendre transform α(q) = ∂[(q − 1)Dq] ∂q , (2) f(α) = q α(q)− (q − 1)Dq. (3) In fractal measures defined on simple fractals, the values of α are usually bounded from above and from below, αmin ≤ α ≤ αmax. When the fractal measure fails to exhibit power law scaling everywhere (say, Pi( ) ∼ e− somewhere), one finds a phase transition in this formalism [8] with one of the edge values αmin or αmax ceasing to exist. When the spectrum of exponents α and their frequency of occurrence f(α) in two problems are the same, we refer to these problems as being in the same universality class. We note that a similar criterion was used to state that different dynamical systems are in the same universality class at their transition to chaos [9], but not many other physical problems yielded to such a stringent test, simply because it is not easy to compute with enough precision the scaling properties of fractal measures. For DLA, such an accurate computation was achieved [10]; in this letter, we report also an accurate computation for experimental viscous fingering patterns, allowing us to test whether the two problems are in the same universality class. The apparent complexity of the viscous fingers is discouraging for any attempt to calculate the harmonic measure reliably. The naive method for computing the harmonic measure for DLA is to “probe” the interface with random walkers and perform frequency statistics [11]. While the statistics at the outer tips is reasonable [12], the deep fjords are visited extremely rarely by the random walkers. Similarly, in viscous fingering experiments, the measure is usually estimated from the velocity of the interface [13]. Obviously, this reveals the outer tips which move with an appreciable velocity, but leaves unmeasured the harmonic measure on the fjords, which almost do not move at all. One has to look for alternative methods. The harmonic measure from iterated conformal maps. The Riemann theorem guarantees that there exists a conformal map from the exterior of the unit circle ω = e to the exterior of our viscous fingering patterns. Having such a conformal map, say Φ(ω), the harmonic measure is simply obtained as 1/|Φ′(ω)| up to normalization. The actual construction of such a map for a given well-developed fractal pattern is, however, far from obvious, and it has never been accomplished before. We will demonstrate now that the method of iterated conformal maps [14, 15] can be adapted to solve this problem in a very efficient way. We use data acquired in experiments, in which air is injected into oil from a central orifice. The oil is confined between two circular glass plates, 288mm in diameter and each 60mm thick [2]. The plates are separated by 0.127mm and are flat to 0.13μm (optically polished to one-quarter of a wavelength). An oil buffer surrounds the plates. The data presented here were obtained with silicone oil with a dynamic viscosity μ = 345mPa s and surface tension σ = 20.9mN/m at 24 ◦C; additional experiments were conducted with silicone oil with a dynamic viscosity μ = 50.8mPa s and surface tension σ = 20.6mN/m at 24 ◦C. The pressure difference ∆p between the injected air and the oil buffer ranged from 0.1 to 1.25 atm. The 1mm diameter central hole through which the air is injected cannot be seen in fig. 1, rather the 22mm diameter central solid circular region (black) is a mask introduced after the images have 260 EUROPHYSICS LETTERS